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Solving Linear Inequalities

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How to solve inequalities in the same way as equations, find the greatest whole number that fits, deal with brackets, fractions and letters on both sides, write an inequality from a fact in words, and solve a double inequality and show it on a number line.

What the exam asks

An inequality uses one of four signs: < (is less than), \leq (is less than or equal to), > (is greater than) or \geq (is greater than or equal to). So x<5 means x can be any number below 5, but not 5 itself. x\leq5 allows 5 as well.

In the exam you are usually asked to solve an inequality such as 4x-3<13, for 2 marks, or to work out the greatest integer that fits. Harder questions, for 3 marks, have x on both sides or a fraction, ask you to write an inequality from a fact about a shape, or give a double inequality, with two signs, to solve and show on a number line.

Solve it like an equation

Solving an inequality means finding all the values of x that make it true. Use the same steps as for an equation: do the same thing to both sides, and keep the inequality sign.

For example, subtract 4 from both sides of x+4>9:

x>5

Test it. x=6 works, because 6+4=10, which is greater than 9. x=5 does not, because 5+4=9, which is not greater than 9.

Your answer is an inequality, such as x>5, not a single number. It describes a whole range of values. Writing x=5, or just 5, loses the last mark.

Worked example 1: A two-step inequality

Solve 6x+5\geq29

  1. Subtract 5 from both sides: 6x\geq24

  2. Your turn. Divide both sides by 6. Write the solved inequality.

    Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

    x\geq24\div6, so x\geq4

  3. Check with x=5: 6\times5+5=35, and 35\geq29 is true.
  4. Answer: x\geq4

When the answer must be an integer

Some questions say "n is an integer", which means n is a whole number, and ask for the greatest possible value of n.

Solve the inequality first, then look carefully at the number at the end. If n<4.6, the greatest integer is 4. If n<4, then 4 is not allowed, so the greatest integer is 3. If n\leq4, it is 4.

Take care with negative numbers. If n<-2.5, the greatest integer is -3, not -2, because -2 is bigger than -2.5

Worked example 2: The greatest integer

n is an integer and 3n+20<6. Work out the greatest possible value of n.

  1. Subtract 20 from both sides: 3n<-14
  2. Divide both sides by 3: n<-\frac{14}{3}, which is n<-4\frac{2}{3}

  3. Your turn. What is the greatest integer that is less than -4\frac{2}{3}?

    The greatest integer less than -4\frac{2}{3} is -5. -4 is too big.

  4. Check: 3\times(-5)+20=5, which is less than 6. But 3\times(-4)+20=8, which is not.
  5. Answer: n=-5

Letters on both sides, brackets and fractions

When x is on both sides, get the x-terms together first. Subtract the smaller x-term from both sides, so the x-term that is left is positive.

If an x-term is taken away, as in x+3>9-2x, add it to both sides instead. Adding 2x to both sides gives 3x+3>9, so 3x>6 and x>2

Expand any brackets before you move anything. 3(x+4) is 3x+12, because the 3 multiplies both terms.

If there is a fraction, multiply every term on both sides by the number underneath. For \frac{x}{4}-1<2, multiply by 4 to get x-4<8. The 1 and the 2 are multiplied by 4 too.

The answer may end up with x on the right, as in 7<x. That means the same as x>7: the sign still points at the smaller side.

Worked example 3: A bracket and x on both sides

Solve 3(x+4)\leq5x-2

  1. Expand the bracket: 3x+12\leq5x-2
  2. The smaller x-term is 3x. Subtract 3x from both sides: 12\leq2x-2
  3. Add 2 to both sides: 14\leq2x

  4. Your turn. Divide both sides by 2. Write the solved inequality with x on the left.

    Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

    7\leq x, which is the same as x\geq7

  5. Check with x=10: 3\times14=42 and 5\times10-2=48. 42\leq48 is true.
  6. Answer: x\geq7

Worked example 4: Clearing a fraction

Solve \frac{2x+1}{3}<x-4

  1. Multiply both sides by 3. The whole of the right-hand side is multiplied: 2x+1<3(x-4)

  2. Your turn. Expand 3(x-4)

    Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

    3(x-4)=3x-12, so 2x+1<3x-12

  3. Subtract 2x from both sides: 1<x-12
  4. Add 12 to both sides: 13<x
  5. Check with x=16: \frac{2\times16+1}{3}=11 and 16-4=12. 11<12 is true.
  6. Answer: x>13

Writing an inequality from words

Some questions give a fact in words, and you write the inequality yourself. At most means \leq, at least means \geq, more than means > and less than means <.

Angle facts work the same way. An acute angle is less than 90^\circ, so use <90. An angle that is not obtuse can be 90^\circ or less, so use \leq90.

Then solve the inequality as usual, and answer the question that was asked. If it asks for the greatest integer, give one number.

Worked example 5: Writing an inequality from a fact

Angle PQR=(3x+15)^\circ. Angle PQR is acute. x is an integer. Work out the greatest possible value of x.

  1. Acute means less than 90^\circ, so 3x+15<90
  2. Subtract 15 from both sides: 3x<75
  3. Divide both sides by 3: x<25

  4. Your turn. x is an integer. What is the greatest value it can be?

    x must be less than 25, so 25 is not allowed. The greatest integer is 24

  5. Check: 3\times24+15=87, which is acute. But 3\times25+15=90, which is a right angle, not acute.
  6. Answer: 24

Double inequalities

A double inequality such as -1<x+2\leq4 has three parts. It says that x+2 is greater than -1 and also less than or equal to 4.

Do the same thing to all three parts. Subtract 2 from each part: -3<x\leq2

If x is an integer, the possible values here are -2, -1, 0, 1 and 2. -3 is not one of them, because the sign next to it is <, so -3 is not included.

Worked example 6: Solving a double inequality

Solve -3\leq x-2<3. Show your answer on a number line.

  1. To leave x on its own in the middle, add 2 to all three parts.

  2. Your turn. What does the left-hand part, -3, become?

    -3+2=-1, so the left-hand part is -1

  3. The right-hand part becomes 3+2=5, so -1\leq x<5
  4. Check with x=4: 4-2=2, and -3\leq2<3 is true.
  5. On the number line, -1 is included, so it gets a filled circle. 5 is not included, so it gets an open circle. Join them with a straight line.
  6. Answer: -1\leq x<5, as shown on the number line.

Common mistakes

  • Giving a number instead of an inequality. If you solve 7x<35, the answer is x<5, not x=5 or 5
  • Multiplying only some of the terms. When you clear a fraction, multiply every term on both sides, not just the one with the fraction.
  • Combining an x-term with a number. 3x+9 is not 12x. Get the x-terms on one side and the numbers on the other.
  • Choosing an integer that is not allowed. If n<5, the greatest integer is 4; if n<-4\frac{2}{3}, the greatest is -5, not -4
  • Guessing with trial and improvement. Trying numbers often gives the wrong value at the end, so write the inequality and solve it step by step.
  • Changing only one part of a double inequality. Do the same thing to all three parts.

Practice questions

More on this topic: Solving Linear Inequalities worksheet with full solutions